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Math Complex Numbers and Parametric MotionIntermediateStarter track

Concept module

Trig Identities from Unit-Circle Geometry

Keep one rotating point and its projections visible so the core trig identities stay tied to geometry instead of detached symbol rules.

Interactive lab

Keep the stage, immediate control feedback, and linked study tools in one working bench.

Unit circle rotation

Keep one rotating point, its x and y projections, and the sine-cosine traces tied to the same angle so the unit circle reads as a live source for both functions.

Drag the point or use the phase slider

Controls

Adjust the live parameters and watch the bench respond.

Angular speed

ω

1 rad/s

Controls how quickly the point rotates around the unit circle.

Starting angle

θ

2.21 rad

Sets where the point starts before the live rotation begins.

Presets

Time

0.00 s / 6.28 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s6.28 s

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 2

The cosine-square and sine-square traces trade size, but their sum stays pinned at 1 because the point never leaves the unit circle.

Graphs

Switch graph views without breaking the live stage and time link.

Squared projections and their sum

Square the current cosine and sine projections over the same rotation so the identity stays visible while the raw signs change.

time: 0 to 6.28value: 0 to 1.15

cos^2(θ)sin^2(θ)cos^2(θ) + sin^2(θ)

Hover or scrub to link the graph back to the stage.time / value

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

ω

Angular speed

1 rad/s

Changes how quickly the point sweeps through the circle without changing the identity itself.

Graph: Cosine and sine tracesGraph: Squared projections and their sumOverlay: Rotation trailOverlay: Angle marker

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Keep the circle and the identity graph visible together.

ObservationPrompt 1 of 2

Graph: Squared projections and their sum

The cosine-square and sine-square traces trade size, but their sum stays pinned at 1 because the point never leaves the unit circle.

Control: Starting angleGraph: Squared projections and their sumOverlay: Projection guidesOverlay: Rotation trailEquation

θ

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

4 visible

Overlay focus

Projection guides

Project the current point back to the x and y axes.

What to notice

The same horizontal and vertical shadows become the quantities being squared on the identity graph.

Why it matters

It keeps the identity attached to the visible geometry.

Control: Starting angleGraph: Cosine and sine tracesGraph: Squared projections and their sumEquation

θ

Challenge mode

Use the unit-circle point, the projection guides, and the squared graph together. This checkpoint is about building a concrete identity case from one live angle instead of quoting the formula cold.

0/1 solved

MatchCore

4 of 5 checks

Three-four-five identity checkpoint

Start in Quadrant I and set the point so cosine is near 0.6 while sine is near 0.8. Keep the squared-projection graph open so the identity line stays in view while you tune the angle.

Graph-linkedGuided start2 hints

Suggested start

Aim for a first-quadrant point a little steeper than 45 degrees.

Matched

Open the Squared projections and their sum graph.

Squared projections and their sum

Matched

Keep the Projection guides visible.

Matched

Keep the Angle marker visible.

Matched

Keep the Rotation trail visible.

Pending

Set the starting angle between about 0.89 and 0.96 rad so the projections land near 0.6 and 0.8.

2.21 rad

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the unit-circle point sits in Quadrant II at theta = 126.62°. The horizontal projection is cos(theta) = -0.6, the vertical projection is sin(theta) = 0.8, and one full turn takes 6.28 s.

Equation detailsDeeper interpretation, notes, and worked variable context.

Distance rule on the unit circle

Any point on the unit circle stays exactly one unit from the origin.

x^{2} + y^{2} = 1

ω

Angular speed 1 rad/s

Pythagorean identity

Substituting x = cos theta and y = sin theta into the unit-circle distance rule gives the core identity.

cos^{2} θ + sin^{2} θ = 1

ω

Angular speed 1 rad/s

θ

Starting angle 2.21 rad

Complementary-angle swap

In the first quadrant, complementary angles swap the horizontal and vertical projections of the same right-triangle geometry.

sin θ = cos (9 0^{\circ} - θ), cos θ = sin (9 0^{\circ} - θ)

θ

Starting angle 2.21 rad

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 1 compact task ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

This bench opened from a setup link. Any new copy will reflect the state you see now.

Current bench

Quadrant II balance preset

This bench still matches one named preset, so the copied link will reopen that same starting point along with the current graph, overlays, and inspect context.

Open default bench

Featured setups

Identity balance

Open a first-quadrant point with the squared-projection graph already in view.

Quadrant II balance

Move the point left of the y-axis and watch the squared sum stay fixed anyway.

Saved setups

Premium keeps named exact-state study setups in your account while stable concept links stay public below.

Checking saved setup access.

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Copy current setup

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Stable links

Starter track

Step 4 of 60 / 6 complete

Complex and Parametric Motion

Earlier steps still set up Trig Identities from Unit-Circle Geometry.

1. Complex Numbers on the Plane2. Unit Circle / Sine and Cosine from Rotation3. Polar Coordinates / Radius and Angle4. Trig Identities from Unit-Circle Geometry+2 more steps

Previous step: Polar Coordinates / Radius and Angle.

Start from track beginning

Short explanation

What the system is doing

Trig identities land more honestly when they stay on the same unit-circle bench as the rotating point. This page keeps the point, its projections, and a squared-projection graph visible together so the identities come from geometry instead of looking like detached symbol tricks.

On the unit circle the radius is always 1, so every live point satisfies x^2 + y^2 = 1. Once the same point is read as (cos theta, sin theta), the Pythagorean identity is forced by the picture. Complementary-angle swaps are geometric too: in the first quadrant, swapping the horizontal and vertical shadows swaps cosine and sine.

Key ideas

01Because the unit-circle radius is 1, every live point satisfies x^2 + y^2 = 1.

02Replacing x with cos theta and y with sin theta gives cos^2 theta + sin^2 theta = 1.

03Raw sine and cosine signs can change across quadrants, but the squared sum stays fixed because the distance from the origin does not change.

04Complementary angles swap the horizontal and vertical projections in the first quadrant, which is why sine and cosine trade places.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use the identity-balance preset so the point and the squared graph stay aligned.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans

Frozen valuesUsing frozen parameters

For the identity-balance preset, what does the unit-circle point say about the core identity?

θ

Starting angle

53.1 °

1. Read the live projections

At about 53.1 deg, the unit-circle point is close to (0.60, 0.80), so cos theta is about 0.60 and sin theta is about 0.80.

2. Square the same two projections

That gives cos^2 theta \approx 0.36 and sin^2 theta \approx 0.64.

3. Add the squared pieces

The squared-projection graph shows the same result numerically: 0.36 + 0.64 = 1.00.

Identity check

\cos^2\theta + \sin^2\theta = 1

The identity is not a separate algebra trick. It is the distance formula for one point that never leaves the unit circle.

Common misconception

Trig identities are mostly symbolic algebra, so the unit-circle picture is optional once the formulas are memorized.

The identities come from the same geometry as the live projections.

If the point stays on the unit circle, the squared projections must still add to 1 even when the raw signs change.

Mini challenge

Move the point from Quadrant I into Quadrant II without changing its distance from the origin. What has to stay true about the squared projections?

Prediction prompt

Decide whether the sign change should affect the squared sum or only the raw cosine sign.

Check your reasoning

The squared sum still has to stay equal to 1.

Crossing into Quadrant II flips the raw cosine sign, but the point is still on the same unit circle, so the distance formula does not change.

Quick test

Misconception check

Question 1 of 2

Answer from the live point and the squared graph.

What is the cleanest geometric source of cos^2 theta + sin^2 theta = 1 on this bench?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a unit circle with one rotating point, horizontal and vertical projection guides, an angle marker, a quadrant sign map, and graphs that compare the raw projections with their squared values.

Graph summary

One graph shows cosine and sine changing over time, and a second graph shows cosine squared, sine squared, and their sum staying fixed at 1.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Use the same plane to recover full angles from ratios without dropping the quadrantComplex Numbers and Parametric Motion

Trig Identities from Unit-Circle Geometry

See each variable before you move it.

Projection guides

Three-four-five identity checkpoint

Complex and Parametric Motion

What the system is doing

Read the full frozen walkthrough.

For the identity-balance preset, what does the unit-circle point say about the core identity?

What is the cleanest geometric source of cos^2 theta + sin^2 theta = 1 on this bench?

Keep moving through the concept map.

Inverse Trig / Angle from Ratio

Parametric Curves / Motion from Equations

Derivative as Slope / Local Rate of Change